**4. Dual symbols for space-group operations**

As mentioned in the previous section, the geometric interpretation of symmetry operations is vital for many crystallographic topics. But there are some disadvantages in the derivation and application of the pioneers symmetry operation symbols (Fischer, Koch in ITA83). In the published procedure, intended mainly for manual calculations and for a link between symmetry matrices and the diagrams of symmetry elements, there is a lack of conventions needed to cast this approach into a programmable algorithm and to obtain unique relations: given symmetry operation the operation symbol. In the literature concerning this topic individual conventions are frequently added into specified algorithms [8,9], which causes that the derived symbols are unique, but not necessary agree with ITA83 symbols. Special difficulties arise in deriving the orientation-location part of a symmetry operation. A systematic analysis of ITA83 symbols resulted in extracting 6 conventions necessary for automatic generation of the orientation-location parts in an unambiguous way [14]. The novel approach presented in that publication contains the derivation of a *point closest to origin* **x**c as an alternative to solving an indeterminate set of equations by the pseudoinversion matrices. But a methodological advance connected with the unambiguity of **x**c was next lost in the standardization step, in which the obtained result was transformed into a conventional point on a geometric element. The inconveniences of classical symbols can be summarized as follows:

254 Recent Advances in Crystallography

For example the matrix equation

or in post-calculation standardization.

even for coordinate triplets taken from ITA83 [18].

�

1. *x*, -*x* + 1/3, -*x* + 1/6 if *x* is treated as the parameter, 2. –*y* + 1/3, *y*, *y* – 1/6 if *y* is selected as the parameter, 3. –*z* + 1/6, *z* + 1/6, *z* if *z* represents the variable parameter.

**3.4. Pseudo-inverse and the point closest to origin** 

and specification in the geometric descriptions.

**4. Dual symbols for space-group operations** 

001� 1� 0 0 010

gives three solutions presented here as the orientation-location parts:

� � � � � ���

calculation.

In both cases the set of equations is indeterminate, since det(**W** - **I**) in (12) or det(**W2** - **I**) in (13) are equal zero. This leads to parametric solutions, which forms depend on the mode of

Although the procedure may also be applied to cases where space groups are given in nonconventional descriptions, there are situations where it is difficult to obtain a unique result

Of course, these solutions describe the same symmetry axis, but differ in a selected parameter, in a positive direction and in a location point. Unique solutions require special conventions in programmable algorithms, based for example on the *row echelon forms* [9, 27],

Another possibility originates from the linear algebra and from the concept of *pseudo-inverse* matrices. This mathematical formulation allows obtaining single points **x**F from equations (12) and (13). Such unique points have simple interpretation; they represent points from the linear or planar sets closest to origin. The derivation of the pseudo-inverse matrix (**W**-**I**)+ is rather cumbersome, but this idea inspires the new point of view on the location derivation

As mentioned in the previous section, the geometric interpretation of symmetry operations is vital for many crystallographic topics. But there are some disadvantages in the derivation and application of the pioneers symmetry operation symbols (Fischer, Koch in ITA83). In the published procedure, intended mainly for manual calculations and for a link between symmetry matrices and the diagrams of symmetry elements, there is a lack of conventions needed to cast this approach into a programmable algorithm and to obtain unique relations: given symmetry operation the operation symbol. In the literature concerning this topic individual conventions are frequently added into specified algorithms [8,9], which causes that the derived symbols are unique, but not necessary agree with ITA83 symbols. Special difficulties arise in deriving the orientation-location part of a symmetry operation. A

1/6 1/3 �1/6

��� � � � �


Some of the listed problems are immanently involved with the specific form of the orientation-location part and they should be overcome or at least simplified by developing a new symbol of symmetry operation. The problems (ii) and partially (iii) and (iv) disappear, if the orientation of a geometric element is presented in the form of the orthogonal lattice splitting (*uvw*)[*hkl*]. Advantages of using splitting indices for the characterization of point operations were described in [23]. The benefits for space operations should be even greater. The conventions for a unique description of a geometric element location in (i) have no meaning for **x**c. Moreover, by removing the *a*, *b*, *c*, *g* designations of glide planes, a new symbol will be more consistent.

The splitting indices describe the orientation of the symmetry axes or the symmetry planes in the same way as [*uvw*](*hkl*). They, together with a locating point on the geometric element, fully correspond to the orientation-location parts. More precisely, the informative content of this construction exceeds the analogous information stored classically. Lattice rows [*uvw*] and lattice planes (*hkl*) are orthogonal to each other and exchangeable in the reciprocal space. Specifications of the same lattice direction in direct and reciprocal spaces better characterize the axis system and simplify the crystallographic calculations, that is calculations in non-Cartesian systems. While two point operations given by coordinate triplets �� � �� � and ̅ �� � �̅� �� �̅ lead to a common geometric symbol 2 0,*y*,0, the splitting indices are [010](010) in the former case and [010](1�20) in the latter.

The lattice perpendicularities revealed by splitting indices are also important in the orthogonal splitting of the translational part **w** of a space-group operation. The **w** components are parallel/orthogonal to [*uvw*] and also to (*hkl*), what was symbolically drawn in Figure 1.

**Figure 1.** Origin - central lattice point of a dual symbol. Each crystallographic point operation (excluding identity and inversion) orthogonally splits the lattice into the lattice rows [*uvw*] and the lattice planes (*hkl*). The decomposition of a translation **w** in the space-group operations is also based on the lattice perpendicular property.

Let **x**c denotes a special point, the point from set **x**F closest to the coordinate system origin. Such points may be found on any geometric element (Fig.2). Thus, every 4x4 symmetry matrix **W**, excluding only identity operation and pure translations, can be characterized by **x**c. Vector **x**c is always perpendicular to the geometric element and to the intrinsic part **w**g, if not zero. Two vectors separated by the asterisk, that is **w**g\* **x**c give complete geometric information about a space-group operation, assuming that the point –group operation was characterized by the dual symbol. For this reason, the new geometric symbol of a spacesymmetry operation is also called the *dual symbol*, like in the case of point operations [18]. This should not lead to confusion, since the information concerning the point operation is separated from the characteristics involved with the space-symmetry operation. Such approach reflects the analogy to the algebraic difference between the point operation **W** and the space operation (**W**,**w**). Moreover, the perpendicularity between the lattice row and the lattice planes expressed by splitting indices is reflected also in the pair **w**g, **x**c. In consequence, the six components of two vectors **w**g\* **x**c may be presented only by four simple ratios, since in every case **w**g or **x**c is parallel to the integer vector [*uvw*]. Thus, the dual symbol takes the form

$$\begin{aligned} & \pm n^{\pm} \left[ \iota \upsilon \upsilon w \right] \left( hkl \right) r\_1 \ast r\_2, r\_3, r\_4 \text{ for rotations and rotations} \\ & \text{and } m \left[ \iota \upsilon \upsilon w \right] \left( hkl \right) r\_1, r\_2, r\_3 \ast r\_4 \text{ for reflections,} \end{aligned} \tag{14}$$

where: the minus sign before *n* or a bar over *n* is the 'inversion sign' = det(**W**) *n* – axis symbol, order of **W**p, *m* – mirror plane (*n* =2, det(**W**) = -1), [*uvw*] – symmetry axis direction or a normal to the reflection plane, [*hkl*] – reflection plane or plane perpendicular to the direction of symmetry axis, *r*1 \* *r*2, *r*3, *r*4 – specification of vectors **w**g = (*r*1*u*, *r*1*v*, *r*1*w*) and **x**c = (*r*2, *r*3, *r*4), *r*1, *r*2, *r*3 \* *r*4 – specification of vectors **w**g = (*r*1, *r*2, *r*3) and **x**c = (*r*4*u*, *r*4*v*, *r*4*w*).

256 Recent Advances in Crystallography

the lattice perpendicular property.

symbol takes the form

 

 

*n uvw hkl r r r r m uvw hkl r r r r*

1 234 1 2 3 4

and , , \* for reflections,

\* , , for rotations and rotoinversions

(14)

**Figure 1.** Origin - central lattice point of a dual symbol. Each crystallographic point operation (excluding identity and inversion) orthogonally splits the lattice into the lattice rows [*uvw*] and the lattice planes (*hkl*). The decomposition of a translation **w** in the space-group operations is also based on

Let **x**c denotes a special point, the point from set **x**F closest to the coordinate system origin. Such points may be found on any geometric element (Fig.2). Thus, every 4x4 symmetry matrix **W**, excluding only identity operation and pure translations, can be characterized by **x**c. Vector **x**c is always perpendicular to the geometric element and to the intrinsic part **w**g, if not zero. Two vectors separated by the asterisk, that is **w**g\* **x**c give complete geometric information about a space-group operation, assuming that the point –group operation was characterized by the dual symbol. For this reason, the new geometric symbol of a spacesymmetry operation is also called the *dual symbol*, like in the case of point operations [18]. This should not lead to confusion, since the information concerning the point operation is separated from the characteristics involved with the space-symmetry operation. Such approach reflects the analogy to the algebraic difference between the point operation **W** and the space operation (**W**,**w**). Moreover, the perpendicularity between the lattice row and the lattice planes expressed by splitting indices is reflected also in the pair **w**g, **x**c. In consequence, the six components of two vectors **w**g\* **x**c may be presented only by four simple ratios, since in every case **w**g or **x**c is parallel to the integer vector [*uvw*]. Thus, the dual

**Figure 2.** Characteristic points of geometric elements in general orientation. For a symmetry axis they defined the intersection of an axis with the basal planes. A symmetry plane is located by its intersection with the coordinate axes. Alternative fixing of geometric element may be based on the unique point **x**<sup>c</sup> closest to the origin, schematically presented by open circles.

In most cases such complete form can be further reduced. Zero vectors are omitted. If the lattice row indices are equal to the lattice plane indices, a typical situation for a conventional description of space groups, the latter does not need to be specified. Generally, the indices (*hkl*) in dual symbols are related with the indices [*uvw*] by the lattice metric. Since nonorthogonal symmetry matrices occur only in the oriented hexagonal system, the *h* symbol may designate a unique transformation matrix from [*uwv*] to (*hkl*) according to the scheme:

$$[\![uvw]\!]h \to \begin{pmatrix} 2 & \overline{1} & 0 \\ \overline{1} & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix} = \begin{pmatrix} h \\ k \\ l \end{pmatrix},\tag{15}$$

where the matrix elements are scaled to integers and the 3,3 element is arbitrarily set. The common divisor must be removed from the symbol (*hkl*), if present.

It may be surprising that the dual symbols do not contain the specification of an inversion point in the case of rotoinversion. This can be explained by extending the interpretation of **w**g also on such operations. Vector **w**g 'measures' the intrinsic transformation of the point **x**<sup>c</sup> into another point, both on a geometric element

$$\mathbf{w}\_{\mathbf{g}} = (\mathbf{W}, \mathbf{w})\mathbf{x}\_{\mathbf{c}} - \mathbf{x}\_{\mathbf{c}}.\tag{16}$$

Thus, **w**g describes a classical screw/glide vector and a pure inversion of the point **x**c. This means that

$$\mathbf{x\_{inv}} = \mathbf{x\_c} + \mathbf{w\_g}/2\tag{17}$$

so the calculation and specification of **w**g is reasonable for all operation types. The relations and remarks may be better explained by a few examples.

#### **Example 1**

$$\text{The symmetry matrix} \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}, \begin{bmatrix} 0.5 \\ 0 \\ 0.5 \end{bmatrix} \text{is described as } \vec{4}^{\perp} \text{ [001] } \mathbb{M}^{\perp} \text{ \upharpoonright } \mathbb{M}.$$

The obtained dual symbols contain the following data. The matrix describes a fourfold rotoinversion operation. Its geometric element is oriented along direction [001], which is orthogonal to the (001) plane. The point on the axis closest to origin **x**c has coordinates: ¼,- ¼,0. Mapping **x**c is reduced to a pure inversion and generates the shift ½[0,0,1]. Thus, the inversion point is at ¼,-¼,¼, the middle between **x**c and its image.

#### **Example 2**

The matrix � 010 100 001 �,�0.5 0.5 0 � is described as m[1-10] ½,½,0\*.

In this case the matrix describes a glide symmetry operation. The symmetry plane has Miller indices (1-10) and goes through the origin. The glide vector is (½,½,0).

#### **Example 3**

The matrix � �1 1 0 1 10 0 01 �,� <sup>0</sup> 0 0.5 � is described as m[100](2-10) 0,0½\* or shorter as m[100]h 0,0,½\*.

In this case the matrix also describes a glide symmetry operation, but in a hexagonal system. The symmetry plane has Miller indices (21�0) and goes through the origin. The glide vector is (0,0,½).

#### **Example 4**

The matrix � 1� 1� 1� 100 001 �,�0,5 0 0 � occurring in a primitive description of group No 227 is described as 3+ [1�1���(001) \*1/6,1/6,0.

The matrix generates a three-fold rotation. The axis is oriented along [1�1���direction, which is perpendicular to (001) planes and is located by the point 1/6,1/6,0.

It may be noted from the above examples that – introducing the dual symbols – the inconveniences mentioned at the beginning of this section are avoided. The new notation displays the geometric features of a rotation, a rotoinversion, a screw rotation, a reflection, a glide reflection and even an inversion, in a homogeneous way. All such operations correspond to a geometric element (in the form of a single point, line or plane), for which a point closer to the origin always exists, a unique point **x**c. The difference (**W**,**w**)**x**c - **x**c, defines an 'intrinsic transformation'. These two vectors extend the geometric description of a point operation to the description of a space group operation. The symbols are applicable to space groups presented conventionally as well as non-conventionally. The symbols are rather concise, especially in the context of geometric information, which they contain. The use of the splitting indices [*uvw*](*hkl*) seems to be more applicable for space group then for point groups. Crystallographers interpret [*uvw*] as a family of lattice rows and (*hkl*) as a family of planes. For practical purposes there is only one question: is the determination of **x**<sup>c</sup> computationally simple?

#### **5. Derivation of xc points for space-group operations**

258 Recent Advances in Crystallography

means that

**Example 1** 

**Example 2** 

The matrix �

**Example 3** 

The matrix �

(0,0,½).

**Example 4** 

The matrix �

The symmetry matrix �

010 100 001

�1 1 0 1 10 0 01

> 1� 1� 1� 100 001

described as 3+ [1�1���(001) \*1/6,1/6,0.

�,� 0.5 0.5 0

> �,� 0 0 0.5

�,� 0,5 0 0

into another point, both on a geometric element

and remarks may be better explained by a few examples.

010 �1 0 0 0 0 �1

�,� 0.5 0 0.5

inversion point is at ¼,-¼,¼, the middle between **x**c and its image.

indices (1-10) and goes through the origin. The glide vector is (½,½,0).

is perpendicular to (001) planes and is located by the point 1/6,1/6,0.

**w**g also on such operations. Vector **w**g 'measures' the intrinsic transformation of the point **x**<sup>c</sup>

Thus, **w**g describes a classical screw/glide vector and a pure inversion of the point **x**c. This

so the calculation and specification of **w**g is reasonable for all operation types. The relations

The obtained dual symbols contain the following data. The matrix describes a fourfold rotoinversion operation. Its geometric element is oriented along direction [001], which is orthogonal to the (001) plane. The point on the axis closest to origin **x**c has coordinates: ¼,- ¼,0. Mapping **x**c is reduced to a pure inversion and generates the shift ½[0,0,1]. Thus, the

� is described as m[1-10] ½,½,0\*.

In this case the matrix describes a glide symmetry operation. The symmetry plane has Miller

In this case the matrix also describes a glide symmetry operation, but in a hexagonal system. The symmetry plane has Miller indices (21�0) and goes through the origin. The glide vector is

The matrix generates a three-fold rotation. The axis is oriented along [1�1���direction, which

, – . **w Wwx x g cc** (16)

/ **inv c g x xw2** (17)

� is described as 4�+ [001] ½ \* ¼,-¼,0.

� is described as m[100](2-10) 0,0½\* or shorter as m[100]h 0,0,½\*.

� occurring in a primitive description of group No 227 is

Point **x**c was geometrically defined as the point on the geometric element closest to the origin or equivalently, as the shortest position vector which tail is fixed at the origin and the head ends on the geometric element. Such definition is rather descriptive and not much useful for practical purposes. Generally, the derivation must be carried out in a non-Cartesian system for which the metric is not known. The splitting indices of the matrix part **W** make a geometric concept based on which the quantitative components of **x**c may be calculated from the translation part **w** of a space-group operation. For any **W** (excluding identity or inversion operations) the indices of the orthogonal lattice splitting orient the reflection plane by (*hkl*) indices and also orient line [*uvw*] in space perpendicular to this plane or orient the symmetry axis [*uvw*] and specify some plane orthogonal to it. If **w** = **0**, the direction [*uvw*] and the plane (*hkl*) intersect in the coordinate system origin O, like in Fig. 1. In this situation the head and the tail of **x**c coincide with O, as for point operations. But if **w** ≠ **0**, the head of **x**<sup>c</sup> may move along line [*uvw*] for reflections or may move on (*hkl*) plane for any type of rotations. Thus, the construction based on the lattice orthogonality given by the splitting indices enables finding **x**c without complete metric information. The meaning of **x**c for the inversion operation is obvious, since a geometric element in this situation is reduced to one point and for a pure translation **x**c =0.

The key to finding **x**c on the (*hkl*) plane or on the [*uvw*] line is the orbit, a set of points interrelated by the analysed operation. Contrary to general operation (**W**,**w**), its reduced version (**W**,**w**l) leads to the cyclic groups and to a limited number of equivalent points in space. If an arbitrary point is located on direction [*uvw*] for a symmetry plane, or on the (*hkl*) plane for a symmetry axis, then the centre of gravity of the orbit generated by this point and (**W**,**w**l) defines **x**c. Thus, **x**c describes the shift of the orbit generated by (**W**,**w**l) and points closest to the origin for all (**W**,**w**l)n operations. Certainly, for **w**l = **0** the gravity centre coincides with the origin. Systematic derivation of **x**c for all types of space-group operations is clearer with the help of sketches presented in Figure 3. The intrinsic translation parts **w**l





260 Recent Advances in Crystallography

for the representation of translation

Point **x**c = (O1������ + O2������)/2 = **0**

1. Vector to any point, O1������ = **x**

1. Vector to any point, O1������ = **x**

1. Vector to any point, O1������ = **x** 2. Vector to its image, O2������ = **Wx**+**w**<sup>l</sup>

**Figures 3c and 3d** 

2. Vector to its image, O2������ = **Wx**+**w** = -**x**+**w** 

2. Vector to its image, O2������ = **Wx**+**w**<sup>l</sup> = -**x**+**w**<sup>l</sup>

The new position of inversion point **x**c = (O1������ + O2������)/2 = **w**/2

1. Vector to the special point, O1������ = **x**= **- w**/2 2. Vector to its image, O2������ = **Wx**+**w** = - **w**/2

**Figure 3a** 

**Figure 3b** 

hence

points.

a reflection

**Figures 3e and 3f** 

and the projection planes (*hkl*) are treated as already known. The considered orbits are oneor two-dimensional, assuming the origin is located in the fixed point of an rotoinversion operation. Not all points on the orbit are necessary to derive **x**c. For an orbit in the form of square its centre is determined by points in two opposite vertices. The centre of the hexagon may be reached by rotating one side by 60°. In final formulae the rotation sense is guaranteed by the presence of **W,** but in geometric constructions the rotation sense must be

Identity operation (**I**,**0**) can be interpreted as the reduced operation of a pure translation (**I**,**w**). Its geometric element contains all points. Point (0,0,0) is the closest to the origin. Thus, **x**c is defined as the zero vector, not dependent on **w**. Another, somewhat artificial but more consistent with the rest of figures approach, is based on the selection of specific point **-w**/2

The origin is located at the inversion point. The combination of the inversion with the translation is equivalent to shifting the inversion point by a half of the translation vector,

The orbit generated by a reduced two-fold operation or reduced reflection contains two

The shortest vector **x**c = (O1������ + O2������)/2 = **w**l/2 lies on (*hkl*) for a two-fold rotation or on [*uvw*] for

In the case of four–fold rotation or four-fold rotoinversion the calculations are as follows:

3. Vector to the image of image, O3������ = **W**(**Wx**+**w**l **)+ w**l = -**x** +**Ww**l+**w**l, since in this case **W**2**x** = -**x**

taken into account. A systematic derivation of the formulae for **x**c is followed below.

**Figure 3.** Sketches for derivation points **x**c based on any point **x** and its image(s). On all drawings the projection plane is specified, the origin and **x**c are marked as large filled and empty circles, respectively. Other explanations are given in the text.

The centre of gravity for all symmetry equivalent points is the same as for two points of opposite vertices. Thus, the new position of the rotation point is **x**c = (O1������ + O3������)/2 = (**Ww**l+**w**l)/2

#### **Figures 3g and 3j**

In the case of three–fold rotation or six-fold rotoinversion, the location vectors from the origin to the vertices of equilateral triangle are given by any point **x** and its two images:


The centre of gravity determines the new position of the rotation point

$$\mathbf{x}\_{\overline{\mathbf{x}}} = (\overline{\mathbf{O}1} + \overline{\mathbf{O}2} + \overline{\mathbf{O}3})/\overline{\mathbf{3}} = (\mathbf{W}\mathbf{w} + 2\mathbf{w})/\overline{\mathbf{3}}, \text{ since } \mathbf{W}^2\mathbf{x} + \mathbf{W}\mathbf{x} + \mathbf{x} = \mathbf{0}.$$

#### **Figures 3h and 3i**

In the case of six–fold rotation or three-fold rotoinversion, the location vectors from the origin to the vertices of regular hexagon are given by any point **x** and its consecutive images. But a simpler way may be based on a rotation of a hexagon side by 60°


The centre of gravity determines the new position of the rotation point

**x**c = (O1������ +**W**12�����) = **Ww**l, since **W**2**x** - **Wx** + **x** = **0,** if **W** describe rotation by 60°.

The obtained results are summarized in Table 4.


\* erroneously stated in [18].

**Table 4.** Formulae for calculation points **x**c.

It may be seen from Table 4 that having classified the matrix part **W** and having decomposed translation part **w**, the derivation of **x**c for any space-group operation is unique and extremely simple. Calculations are based on the standard matrix arithmetic: multiplication of a matrix by a vector, adding two vectors or multiplying a vector by a constant. The geometric interpretation of specified rules as well as their explanation is also simple. A comparison of derived formulae with those described in article [18] reveals an incorrect equation for the axis symbol 6 in the published data. The error was overlooked, since in all conventional descriptions of the space groups, the origin is located on the hexagonal axis.
